Let E be a homotopy commutative ring spectrum, and suppose the ring of cooperations E#E is flat over E# . We wish to address the following question: given a commutative E# -algebra A in E#E-comodules, is there an E# -ring spectrum X with E#X = A as comodule algebras? We will formulate this as a moduli problem, and give a way -- suggested by work of Dwyer, Kan, and Stover -- of dissecting the resulting moduli space as a tower with layers governed by appropriate Andre-Quillen cohomology groups. A special case is A = E#E itself. The final section applies this to discuss the Lubin-Tate or Morava spectra En .

The quest for an obstruction theory to E ∞ ring structures on a spectrum has led a number of authors to the investigation of homology in the category of E ∞ algebras. In this note we present three, apparently very different, constructions and show that when specialized to commutative rings they all agree. (AMS subject classification 55Nxx. Key words: André-Quillen homology, E∞homology).

In the first part of this article, I will state a realization problem for diagrams of structured ring spectra, and in the second, I will discuss the moduli space which parametrizes the problem.

In this document we make good on all the assertions we made in the previous paper “Moduli spaces of commutative ring spectra ” [20] wherein we laid out a theory a moduli spaces and problems for the existence and uniqueness of E∞-ring spectra. In that paper, we discussed the the Hopkins-Miller theorem on the Lubin-Tate or Morava spectra En; in particular, we showed how to prove that the moduli space of all E ∞ ring spectra X so that (En)∗X ∼ = (En)∗En as commutative (En) ∗ algebras had the homotopy type of BG, where G was an appropriate variant of the Morava stabilizer group. This is but one point of view on these results, and the reader should also consult [3], [38], and [41], among others. A point worth reiterating is that the moduli problems here begin with algebra: we have a homology theory E ∗ and a commutative ring A in E∗E comodules and we wish to discuss the homotopy type of the space T M(A) of all E∞-ring spectra so that E∗X ∼ = A. We do not, a priori, assume that T M(A) is non-empty, or even that there is a spectrum X so that E∗X ∼ = A as comodules.

Abstract. We discuss some of the basic ideas of Galois theory for commutative S-algebras originally formulated by John Rognes. We restrict attention to the case of finite Galois groups and to global Galois extensions. We describe parts of the general framework developed by Rognes. Central rôles are played by the notion of strong duality and a trace mapping constructed by Greenlees and May in the context of generalized Tate cohomology. We give some examples where algebraic data on coefficient rings ensures strong topological consequences. We consider the issue of passage from algebraic Galois extensions to topological ones applying obstruction theories of Robinson and Goerss-Hopkins to produce topological models for algebraic Galois extensions and the necessary morphisms of commutative S-algebras. Examples such as the complex K-theory spectrum as a KO-algebra indicate that more exotic phenomena occur in the topological setting. We show how in certain cases topological abelian Galois extensions are classified by the same Harrison groups as algebraic ones and this leads to computable Harrison groups for such spectra. We end by proving an analogue of Hilbert’s theorem 90 for the units associated with a Galois extension.

Abstract. We investigate Γ-cohomology of some commutative cooperation algebras E∗E associated with certain periodic cohomology theories. For KU and E(1), the Adams summand at a prime p, and for KO we show that Γ-cohomology vanishes above degree 1. As these cohomology groups are the obstruction groups in the obstruction theory developed by Alan Robinson we deduce that these spectra admit unique E ∞ structures. As a consequence we obtain an E∞ structure for the connective Adams summand. For the Johnson-Wilson spectrum E(n) with n � 1 we establish the existence of a unique E ∞ structure for its In-adic completion.

The aim of this paper is to give an overview of some of the existing homology theories for commutative (S-)algebras. We do not claim any originality; nor do we pretend to give a complete account. But the results in that field are widely spread in the literature, so for someone who does not actually work in that subject, it can be difficult to trace all the relationships between the different homology theories. The theories we aim to compare are • topological André-Quillen homology • Gamma homology • stable homotopy of Γ-modules • stable homotopy of algebraic theories • the André-Quillen cohomology groups which arise as obstruction groups in the Goerss-Hopkins approach As a comparison between stable homotopy of Γ-modules and stable homotopy of algebraic theories is not explicitly given in the literature, we will give a proof of Theorem 2.1 which says that both homotopy theories are isomorphic

Abstract. We describe some of the basic ideas of Galois theory for commutative S-algebras originally formulated by John Rognes. We restrict attention to the case of finite Galois groups. We describe the general framework developed by Rognes. Central rôles are played by the notion of strong duality and a trace or norm mapping constructed by Greenlees and May in the context of generalized Tate cohomology. We give some examples where algebraic data on coefficient rings ensures strong topological consequences. We consider the issue of passage from algebraic Galois extensions to topological ones applying obstruction theories of Robinson and Goerss-Hopkins to produce topological models for algebraic Galois extensions and the necessary morphisms of commutative S-algebras. Examples such as the Real K-theory spectrum as a KOalgebra indicate that more exotic phenomena occur in the topological setting. We show how in certain cases topological abelian Galois extensions are classified by the same Harrison groups as algebraic ones and this leads to computable Harrison groups for such spectra. We consider the Tate spectrum associated to a G-Galois extension and show that it is trivial, thus generalising an analogous result for algebraic Galois extensions. We end by proving an analogue of Hilbert’s theorem 90 for the units associated with a Galois extension.

He spent 11 years in a succession of postdoctoral positions in Canada, USA and Britain, including two years at the University of Chicago as an L. E. Dickson Instructor and 2 years as an EPSRC Advanced Fellow, before being appointed in 1991 to a Lectureship (and subsequently in 1996 to a Readership) at the University of Glasgow. His research has centred on algebraic topology, especially stable homotopy theory. In particular he has focused on applications of algebra and number theory to complex oriented and periodic cohomology theories (especially K-theory and elliptic cohomology). For a representative overview of his work see [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. In recent years he has been very involved in work on structured ring spectra and related topics, and organised a series of workshops in Glasgow, Bonn and Rosendal (Norway), as well as editing a book based on the first of these [13]. Sarah Whitehouse was awarded a Ph.D. from the University of Warwick in 1994. She spent several years in France, as a Marie-Curie post-doctoral researcher at the Université Paris-Nord and as a Lecturer at the Université d’Artois. She joined the University of Sheffield as a Lecturer in 2002 and was promoted to Senior Lecturer in 2005. Much of her work has involved the algebras of operations or cooperations of generalised cohomology theories [18, 19, 27, 38]. Recently, this has given new results for complex K-theory, cobordism and the Morava Ktheories

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